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In mathematics, an arithmetic surface over a Dedekind domain R {\displaystyle R} with fraction field K {\displaystyle K} is a geometric object having one
Arithmetic_surface
Mathematical theory
context, Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. An arithmetic cycle of codimension
Arakelov_theory
Type of mathematical group
instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group PSL2(ℤ). They, and the hyperbolic surface associated
Arithmetic_Fuchsian_group
Generalization of algebraic variety
X=Spec(ℤ[x,y]/(f)) has a canonical morphism to Specℤ and is called an arithmetic surface. The fibers Xp=X×Spec(ℤ)Spec(𝔽p) are then algebraic curves over
Scheme_(mathematics)
Measure of surface finish or texture
Surface roughness or simply roughness is the quality of a surface that is not smooth and it is hence linked to human (haptic) perception of the surface
Surface_roughness
Soviet mathematician
S. J. Arakelov (1974). "Intersection theory of divisors on an arithmetic surface". Mathematics of the USSR-Izvestiya. 8 (6): 1167–1180. doi:10
Suren_Arakelov
Branch of elementary mathematics
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider
Arithmetic
In mathematics, a Picard modular surface is a complex algebraic surface obtained as an arithmetic quotient of the two-dimensional complex unit ball by
Picard_modular_surface
Branch of algebraic geometry
mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is
Arithmetic_geometry
gave a uniform proof and generalized Ogg's formula to more general arithmetic surfaces. We can also describe ε in terms of the valuation of the j-invariant
Conductor of an elliptic curve
Conductor_of_an_elliptic_curve
Chinese-American mathematician (born 1962)
Number Theory. Zhang's doctoral thesis Positive line bundles on Arithmetic Surfaces (Zhang 1992) proved a Nakai–Moishezon type theorem in intersection
Shou-Wu_Zhang
French mathematician (born 1961)
Jean-Benoît; Charles, François (2022), Quasi-projective and formal-analytic arithmetic surfaces, arXiv:2206.14242, retrieved 2025-12-19 Calegari, Frank; Dimitrov
Jean-Benoît_Bost
Mathematical algorithm
In arithmetic geometry, the Cox–Zucker machine is an algorithm introduced by David A. Cox and Steven Zucker for studying elliptic surfaces. It determines
Cox–Zucker_machine
Property of an algebraic variety
mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Let X
Arithmetic_genus
mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective
Fake_projective_plane
Branch of mathematics
shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works
Geometry
Type of group in group theory
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm {SL}
Arithmetic_group
Concept in mathematics
parametric surfaces, error analysis (mathematics), process control, worst-case analysis of electric circuits, and more. In affine arithmetic, each input
Affine_arithmetic
Type of surface singularity used in algebraic geometry
elliptic singularity of a surface, introduced by Philip Wagreich in 1970, is a surface singularity such that the arithmetic genus of its local ring is 1
Elliptic_singularity
Branch of pure mathematics
branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties
Number_theory
Mathematical theorem
{\displaystyle 1+p_{a}} , where p a {\displaystyle p_{a}} is the arithmetic genus of the surface. For comparison, the Riemann–Roch theorem for a curve states
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Number of "holes" of a surface
number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. The genus of a connected, orientable surface is an integer representing
Genus_(mathematics)
Riemann surfaces with the identical automorphism group (of order 84(14 − 1) = 1092 = 22·3·7·13). The explanation for this phenomenon is arithmetic. Namely
Hurwitz_surface
Branch of mathematics
the real algebraic varieties. Diophantine geometry and, more generally, arithmetic geometry is the study of algebraic varieties over fields that are not
Algebraic_geometry
Form of plant intelligence
Plant arithmetic is a form of plant intelligence whereby plants appear to perform arithmetic operations – a form of number sense in plants. Some such plants
Plant_arithmetic
Concept in algebraic geometry
MR 0833513 Nagata, Masayoshi (1960), "On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1", Mem. Coll. Sci. Univ. Kyoto Ser. A Math
Del_Pezzo_surface
only if the surface is tiled by parallelograms. There exists Veech surfaces whose Veech group is not arithmetic, for example the surface obtained from
Translation_surface
Smooth closed surface with g holes
In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior
Genus_g_surface
Type of smooth complex surface of kodaira dimension 0
theory of K3 surfaces and the arithmetic of symmetric bilinear forms. As a first example of this connection: a complex analytic K3 surface is algebraic
K3_surface
Algebraic variety of dimension two
cubic surfaces, Veronese surface, del Pezzo surfaces, ruled surfaces κ = 0 : K3 surfaces, abelian surfaces, Enriques surfaces, hyperelliptic surfaces κ =
Algebraic_surface
Mathematics
was obtained by Buser and Sarnak. Namely, they exhibited arithmetic hyperbolic Riemann surfaces with systole behaving as a constant times log ( g ) {\displaystyle
Systoles_of_surfaces
Russian mathematician
Iwasawa-Tate theory from 1-dimensional global fields to 2-dimensional arithmetic surfaces such as proper regular models of elliptic curves over global fields
Ivan_Fesenko
Mathematics of varieties with integer coordinates
these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems of fundamental importance in Diophantine geometry
Diophantine_geometry
Chinese mathematician (born 1981)
University working in number theory, arithmetic geometry, and automorphic forms. In particular, his work focuses on arithmetic intersection theory, algebraic
Xinyi_Yuan
(2018). Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics: CIRM Jean-Morlet Chair, Fall 2016. Springer. p. 185
List_of_conjectures
Mathematical model
minimal model over R in the sense of algebraic (or arithmetic) surfaces. This is a regular proper surface over R but is not in general smooth over R or a
Néron_model
Concept in algebraic geometry
2-dimensional schemes (including all arithmetic surfaces) by Lipman (1978). Zariski's method of resolution of singularities for surfaces is to repeatedly alternate
Resolution_of_singularities
Number taken as representative of a list of numbers
its overall position. In mathematics, it most commonly refers to the arithmetic mean, but may also refer to other measures such as other types of mean
Average
Non-singular cubic surface in mathematics
mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by Clebsch (1871) and Klein
Clebsch_surface
Area of mathematics
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields
Arithmetic_topology
Calculating tool
in Roman abacus), and a decimal point can be imagined for fixed-point arithmetic. Any particular abacus design supports multiple methods to perform calculations
Abacus
Algebraic surface in mathematics
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies
Hilbert_modular_variety
{\displaystyle p_{g}-p_{a}} of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending
Irregularity_of_a_surface
ISO collection of international standards
3D areal surface texture. Documents constituting the standard: Part 1: Indication of surface texture Part 2: Terms, definitions and surface texture parameters
ISO_25178
Graduate-level textbooks in mathematics
the Theory of Riemann Surfaces. Edited by Lars V. Ahlfors, Lipman Bers 1971-07-21 430 9780691080819 67 Profinite Groups, Arithmetic, and Geometry. Stephen
Annals_of_Mathematics_Studies
Central computer component that executes instructions
electronic circuitry executes instructions of a computer program, such as arithmetic, logic, controlling, and input/output (I/O) operations. This role contrasts
Central_processing_unit
Fourth planet from the Sun
tenuous atmosphere that is primarily carbon dioxide (CO2). At the average surface level the atmospheric pressure is a few thousandths of Earth's, atmospheric
Mars
Mathematical classification of surfaces
quasi-elliptic surfaces in characteristics two and three. These are surfaces fibred over a curve where the general fibre is a curve of arithmetic genus one
Enriques–Kodaira classification
Enriques–Kodaira_classification
Relation between genus, degree, and dimension of function spaces over surfaces
It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over
Riemann–Roch_theorem
Programmable machine that processes data
machine that can be programmed to automatically carry out sequences of arithmetic or logical operations (computation). Modern digital electronic computers
Computer
Mathematical space with two coordinates
system of polynomial equations. Some mathematical spaces have additional arithmetical structure associated with their points. A vector plane is an affine plane
Two-dimensional_space
Geometric figure which has infinite surface area but finite volume
Torricelli's trumpet) is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian idea that the
Gabriel's_horn
Measurement of small-scale features on surfaces
Surface metrology is the measurement and characterization of surface topography, and is a branch of metrology. Surface primary form, surface fractality
Surface_metrology
Number, approximately 3.14
complex numbers at which exp z is equal to one is then an (imaginary) arithmetic progression of the form: {…,−2πi,0,2πi,4πi,…}={2πki∣k∈ℤ} and there is
Pi
German mathematician (born 1958)
results in various directions, such as non-torsion sheaves (1986), arithmetic surfaces (1987), as well as higher-dimensional local fields (with Wingberg
Christopher_Deninger
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Geometric space with four dimensions
August Ferdinand Möbius in Der barycentrische Calcul published 1827. An arithmetic of four spatial dimensions, called quaternions, was defined by William
Four-dimensional_space
Land area where water converges to a common outlet
A drainage basin is an area of land in which all flowing surface water converges to a single point, such as a river mouth, or flows into another body
Drainage_basin
In mathematics, a Riemann surface
mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 {\displaystyle
Bolza_surface
Algebraic structure with addition, multiplication, and division
order, are most directly accessible using modular arithmetic. For a fixed positive integer n, arithmetic "modulo n" means to work with the numbers Z/nZ =
Field_(mathematics)
Chinese mathematician
July 1989) is a Chinese mathematician specialising in number theory and arithmetic geometry. She is a professor of mathematics at the University of California
Yunqing_Tang
Geometric model of the physical space
geodesic on a surface deriving the first analytical geodesic equation, and later introduced the first set of intrinsic coordinate systems on a surface, beginning
Three-dimensional_space
Distance from the Earth surface to a point near its center
RE) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid (an oblate ellipsoid)
Earth_radius
Perimeter of a circle or ellipse
Almkvist, Gert; Berndt, Bruce (1988), "Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary", American Mathematical
Circumference
Temperature at the boundary layer of a fluid undergoing convection
convection boundary layer. It is calculated as the arithmetic mean of the temperature at the surface of the solid boundary wall (Tw) and the free-stream
Film_temperature
Property of a mathematical space
specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D)
Dimension
Operating system
supercomputer and had a nominal performance of 200 megaflops on double precision arithmetic and double that on single precision. The SuperSPARC processors ran at
Meiko_Scientific
Field of knowledge
Euclid's Elements. Mathematics was primarily divided into geometry and arithmetic until the 16th and 17th centuries, when algebra and infinitesimal calculus
Mathematics
Storage of digital data readable by computers, using memory and storage devices
unit and the arithmetic logic unit (ALU). The former controls the flow of data between the CPU and memory, while the latter performs arithmetic and logical
Computer_data_storage
Property of algebraic varieties and complex manifolds
extended by birational invariance. Genus (mathematics) Arithmetic genus Invariants of surfaces Danilov & Shokurov (1998), p. 53 P. Griffiths; J. Harris
Geometric_genus
Also known as higher arithmetic, another name for number theory. Arithmetic algebraic geometry See arithmetic geometry. Arithmetic combinatorics the study
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Can be constructed by light shining through a globe onto a developable surface. 360 video projection List of national coordinate reference systems Snake
List_of_map_projections
Matrix group
More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion
Congruence_subgroup
Type of non-Euclidean geometry
also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in
Hyperbolic_geometry
Reference surface for vertical positions
navigation, vertical datum or altimetric datum is a reference coordinate surface used for vertical positions, such as the elevations of Earth-bound features
Vertical_datum
Straight line segment that passes through the centre of a circle
Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex
Diameter
Conjecture on zeros of the zeta function
every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connected equidimensional arithmetic scheme
Riemann_hypothesis
Surface described by a 4th-degree polynomial
said to be an arithmetic quartic surface. Dupin cyclides The Fermat quartic, given by x4 + y4 + z4 + w4 =0 (an example of a K3 surface). More generally
Quartic_surface
American mathematician (1949–2019)
67 (1): 3–20. MR 0949269. Saper, Leslie; Stern, Mark L2-cohomology of arithmetic varieties, Annals of Mathematics (2) 132 (1990), no. 1, 1–69. MR 1059935
Steven_Zucker
Measurement of population size per unit area or unit volume
area of Puerto Rico, 8,868 square kilometres (3,424 sq mi). Although the arithmetic density is the most common way of measuring population density, several
Population_density
1743 arithmetic book by Thomas Dilworth
Assistant, Being a Compendium of Arithmetic both Practical and Theoretical was an early and popular English arithmetic textbook, written by Thomas Dilworth
The Schoolmaster's Assistant, Being a Compendium of Arithmetic Both Practical and Theoretical
The_Schoolmaster's_Assistant,_Being_a_Compendium_of_Arithmetic_Both_Practical_and_Theoretical
Method of drawing geometric objects
is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots
Straightedge and compass construction
Straightedge_and_compass_construction
Proof that only uses basic techniques
, what logicians call an arithmetical statement) can be proved in elementary arithmetic." The form of elementary arithmetic referred to in this conjecture
Elementary_proof
Riemann–Roch theorem Arithmetic Riemann–Roch theorem Riemann–Roch theorem for smooth manifolds Riemann–Roch theorem for surfaces Grothendieck–Hirzebruch–Riemann–Roch
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Mathematics award
Infinitely Small Quantities in Leibniz's Mathematics: The Case of his Arithmetical Quadrature of Conic Sections and Related Curves". In Goldenbaum, Ursula;
Fields_Medal
Country in South Asia
contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. Trigonometry was further advanced in India, and the modern
India
closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for taxation, commerce, trade, and in astronomy
History_of_mathematics
Topics referred to by the same term
the n-dimensional analogue of a square and a cube Hyperoperation, an arithmetic operation beyond exponentiation Hyperplane, a subspace whose dimension
Hyper
Indian mathematician (c. 598–c. 668)
al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as Algorithmi de
Brahmagupta
German polymath and scholar (1777–1855)
Gauss's inequality. Further, he was instrumental in the development of the arithmetic–geometric mean. He introduced the normal or Gaussian distribution in probability
Carl_Friedrich_Gauss
American actress (born 1956)
later, she co-produced and co-starred in the Holocaust drama The Devil's Arithmetic. Together with her fellow producers, Rogers received a Daytime Emmy Award
Mimi_Rogers
Property of magnitude or multitude
Psychology, 40, 235–252. Newton, I. (1728/1967). Universal Arithmetic: Or, a Treatise of Arithmetical Composition and Resolution. In D.T. Whiteside (Ed.), The
Quantity
Symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
Spread of Hindu–Arabic Numerals in the European Tradition of Practical Arithmetic: a Socio-Economic Perspective (13th–16th centuries) (PhD thesis). University
Arabic_numerals
Topics referred to by the same term
for the maintenance of software applications Arithmetic shift left, an operation implementing an arithmetic shift Above sea level, an altitude measurement
ASL_(disambiguation)
General-purpose programming language
executable form was not entirely machine language; rather, floating-point arithmetic, sub-scripting, input/output, and function references were interpreted
Fortran
Product of a number by itself
for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that
Square_(algebra)
Three Riemann surfaces with same symmetry
a unique Hurwitz surface, respectively the Klein quartic and the Macbeath surface). The explanation for this phenomenon is arithmetic. Namely, in the ring
First_Hurwitz_triplet
Topics referred to by the same term
Implicit-association test, in social psychology Implicit bit, in floating-point arithmetic Implicit learning, in learning psychology Implicit memory, in long-term
Implicit
Solid with six equal square faces
1090/S1061-0022-10-01105-2. Khattar, Dinesh (2008). Guide to Objective Arithmetic (2nd ed.). Pearson Education. p. 377. ISBN 9788131716823. Fuchs, Dmitry;
Cube
Quantity of a three-dimensional space
three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral
Volume
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